![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cmpfii | Structured version Visualization version GIF version |
Description: In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Ref | Expression |
---|---|
cmpfii | ⊢ ((𝐽 ∈ Comp ∧ 𝑋 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑋)) → ∩ 𝑋 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6852 | . . . . 5 ⊢ (Clsd‘𝐽) ∈ V | |
2 | 1 | elpw2 5300 | . . . 4 ⊢ (𝑋 ∈ 𝒫 (Clsd‘𝐽) ↔ 𝑋 ⊆ (Clsd‘𝐽)) |
3 | 2 | biimpri 227 | . . 3 ⊢ (𝑋 ⊆ (Clsd‘𝐽) → 𝑋 ∈ 𝒫 (Clsd‘𝐽)) |
4 | cmptop 22730 | . . . . 5 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top) | |
5 | cmpfi 22743 | . . . . 5 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅))) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝐽 ∈ Comp → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅))) |
7 | 6 | ibi 266 | . . 3 ⊢ (𝐽 ∈ Comp → ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)) |
8 | fveq2 6839 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (fi‘𝑥) = (fi‘𝑋)) | |
9 | 8 | eleq2d 2823 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (∅ ∈ (fi‘𝑥) ↔ ∅ ∈ (fi‘𝑋))) |
10 | 9 | notbid 317 | . . . . 5 ⊢ (𝑥 = 𝑋 → (¬ ∅ ∈ (fi‘𝑥) ↔ ¬ ∅ ∈ (fi‘𝑋))) |
11 | inteq 4908 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ∩ 𝑥 = ∩ 𝑋) | |
12 | 11 | neeq1d 3001 | . . . . 5 ⊢ (𝑥 = 𝑋 → (∩ 𝑥 ≠ ∅ ↔ ∩ 𝑋 ≠ ∅)) |
13 | 10, 12 | imbi12d 344 | . . . 4 ⊢ (𝑥 = 𝑋 → ((¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅) ↔ (¬ ∅ ∈ (fi‘𝑋) → ∩ 𝑋 ≠ ∅))) |
14 | 13 | rspcva 3577 | . . 3 ⊢ ((𝑋 ∈ 𝒫 (Clsd‘𝐽) ∧ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)) → (¬ ∅ ∈ (fi‘𝑋) → ∩ 𝑋 ≠ ∅)) |
15 | 3, 7, 14 | syl2anr 597 | . 2 ⊢ ((𝐽 ∈ Comp ∧ 𝑋 ⊆ (Clsd‘𝐽)) → (¬ ∅ ∈ (fi‘𝑋) → ∩ 𝑋 ≠ ∅)) |
16 | 15 | 3impia 1117 | 1 ⊢ ((𝐽 ∈ Comp ∧ 𝑋 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑋)) → ∩ 𝑋 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∀wral 3062 ⊆ wss 3908 ∅c0 4280 𝒫 cpw 4558 ∩ cint 4905 ‘cfv 6493 ficfi 9342 Topctop 22226 Clsdccld 22351 Compccmp 22721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-om 7799 df-1o 8408 df-en 8880 df-fin 8883 df-fi 9343 df-top 22227 df-cld 22354 df-cmp 22722 |
This theorem is referenced by: fclscmpi 23364 cmpfiiin 40958 |
Copyright terms: Public domain | W3C validator |